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\centerline{Mathematics 375 -- Probability Theory}
\centerline{Information on Exam 2}
\centerline{November 9, 2011}
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{\it General Information}
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The second midterm exam will be given starting at 7:00pm on Thursday evening 
November 17.  (Class on Friday, November 18 will be cancelled.)  The exam will
cover the material we have discussed since the first exam, 
up to and including the material on joint densities from
Section 5.2 and marginal and conditional distributions from Section 5.3.  
(In other words, this is the material from Problem Sets 5 - 8.)  
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\noindent
{\it However ...:  An Important Note} 
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Because of the cumulative nature of this 
subject, some questions may also draw on material 
concerning counting techniques, general facts about independence
of events in a sample space, discrete random variables, 
probability functions, expected values, etc. from 
Exam 1.  See the sample questions below for some examples.   
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\noindent
{\it Format and Groundrules}
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This will be a {\it closed book} exam, but you may 
prepare {\it one side of a $3\times 5$ inch index card}
with formulas and any other information you want to include, 
bring it to the exam and consult it at any time.  
I will provide a copy of the table for normal
probabilities from the text.  Calculators allowed.  No
cell phones, I-pods, pagers or other electronic devices 
allowed.  
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{\it Topics}
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\noindent
The topics to be covered are:
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\item{1)}  The hypergeometric and Poisson discrete probability 
distributions -- Sections 3.7 and 3.8.
\item{2)}  Continuous random variables and probability distributions,
cdfs and pdfs -- Section 4.2
\item{3)}  Expected values for continuous random variables -- Section 4.3
\item{4)}  Uniform, Normal, Gamma, and Beta distributions -- Note
that this includes the definition and properties of the Gamma and 
Beta functions.  Be prepared
for problems like those on the homework where you need to find
the mean and variance of some {\it function of} a random variable
of one of these types, and also for problems where
you might use the geometric or binomial distributions
in combination with one of these -- Sections 4.4 - 4.7
\item{5)}  Moment-generating functions for discrete and continuous random 
variables -- Sections 3.9 and 4.9
\item{6)}  Tchebysheff's Theorem -- Section 4.10
\item{7)}  Joint probability functions, jointly continuous
random variables, joint densities, plus marginal and conditional densities -- Sections 5.2
and 5.3
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{\it Review Session}
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We can schedule an evening review session 
for the exam next week if there is interest. 
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\noindent
{\it Suggested Review Problems}
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From the text: Chapter 3/195, 196, 213,
Chapter 4/160, 161, 164, 165, 167, 176, 181, 185, 
Chapter 5/11, 15
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\noindent
{\it Sample Exam Questions}
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\noindent
Disclaimer:  The following problems represent the approximate
level of difficulty of the problems on the upcoming exam.  The
exam will be about one half of the total length of these problems.
The actual exam problems may differ substantially from these.    
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\noindent
Comments:  Any general formula we have studied for a mean, 
a variance, etc. can be used without comment (i.e. you don't 
need to rederive it in you solution).  Any selection or other
process done ``at random''
includes the requirement that the outcomes of the individual
operations are independent.   
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\noindent
I. Ten college students, exactly four of whom are not of legal age, 
go out to a bar and all order alcoholic beverages.  The waitperson taking 
their order selects 5 students at random from the group 
to ``card'' and must refuse to serve any one who is under legal age.  
What is the probability that exactly 2 students will be refused service?
(Assume, unrealistically perhaps, that each ID shows that student's actual 
age!) 
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II.  The fraction $Y$ of impurities in a tank-car lot of industrial grade
potassium perchlorate is a random variable with probability density function 
$$f(y) = \cases{ c y^2(1-y)^2 & if $0 < y < 1$\cr
                      0 & otherwise\cr}$$
\item{A)}  What is the value of the constant $c$?
\item{B)}  What is the variance of $Y$?
\item{C)}  Find the cumulative distribution function for $Y$.
\item{D)}  Find $P(.1 \le Y \le .25)$.  
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\noindent
III.  The life in years of a certain type of electrical switch
has an exponential distribution with mean $2$ years. 
\item{A)}  What is the probability that a single switch will
fail during the first year after it is installed?
\item{B)}  If 
100 of these switches are installed in different, independently
operating systems, what is the probability that at most 30 fail
during the first year?  (You need {\it not} obtain a single decimal 
approximation to this number; a formula for computing it will suffice.) 
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\noindent
IV.  The weights of a population of miniature poodles are
normally distributed with mean $\mu = 8$kg and standard
deviation $\sigma = .9$kg.  
\item{A)}  What is the probability that
a randomly selected poodle will have weight between 
$7.3$kg and $9.1$kg?
\item{B)}  Let $Y$ be the weight of a poodle from the 
population as a random variable.  What is $E(Y^3)$?  (Note: part credit
will be given for a correct verbal description of a procedure that
could be used to derive this value even if you cannot carry out
the procedure completely.)
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V. Let $Y$ be a random variable
with probability density function 
$$f(y) = \cases{ y e^{-y} & if $y > 0$\cr
                   0 & otherwise\cr}$$
Find the moment-generating function $m_Y(t)$, assuming $t < 1$.
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\noindent
VI.  In a shipment of 10 missiles, there are 3 defectives whose
engines will not fire.  Out of the shipment, 4 missiles are
selected at random. 
\item{A)}  What is the probability that all 4 will fire?
\item{B)}  What is the probability that at most 2 will not fire?
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VII.  In a biomedical research experiment, the survival time $Y$
in weeks of an animal exposed to a certain fatal disease is a random variable
with a gamma distribution with $\alpha = 2$ and $\beta = 6$.
\item{A)} What is the mean survival time of a randomly selected exposed animal?
\item{B)} Without doing any calculations, is it possible that 
$P(|Y - 12| < 12\sqrt{2}) < .7$?
\item{C)} If 6 exposed animals are chosen at random, what is the probability
that at least two survive for 15 or more weeks?
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VIII.  The lifetime of a certain type of small motor is normally distributed
with mean 10 years and standard deviation 2 years.  The manufacturer
replaces free all motors that fail while under warrantee.  But to 
avoid going out of business because of the cost of doing the 
free replacements, the manufacturer must set the warrantee so that
no more than 3\% of the motors that fail will be covered.  What
should the length of the warrantee be?
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IX.  Let $Y_1,Y_2$ be jointly continuous random variables with
joint density:
$$f(y_1,y_2) = \cases{ 30y_1 y_2^2 & if $y_1 -1 \le y_2 \le 1 - y_1$ and $0 \le y_1 \le 1$\cr
                          0 & otherwise\cr}$$
\item{A)}  Find $P(Y_2 > 0)$.  
\item{B)}  Show that the marginal density for $Y_1$ is a beta density with
$\alpha = 2$ and $\beta = 4$.
\item{C)}  What is the conditional density of $Y_2$ given $Y_1 = y_1$?
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